Bolt Falling from elevator ceiling

[cal.queen92]

Homework Statement

An elevator ascends with a constant speed of 2.5m/s. A loose bolt drops from the ceiling of the elevator which is 3.0m above the floor.

a) How long does it take the bolt to fall from the ceiling to the floor of the elevator?
b) What are the displacements of the bolt and the elevator floor in this time?

Homework Equations

constant acceleration (with acceleration equal to gravity at 9.81m/s/s)

The Attempt at a Solution

I managed to get the first part, with t = 1.08 s, but now I am confused with part B. I am not sure what the displacement of the elevator floor really is (I cannot picture it) and I assumed the displacement of the bolt at this time was the 3m that it fell (that's the value of displacement I used to get the time in the first place) SO now I am quite confused.

Can anyone give me a hand?
Thanks!

 

[SammyS]

Homework Statement

An elevator ascends with a constant speed of 2.5m/s. A loose bolt drops from the ceiling of the elevator which is 3.0m above the floor.

a) How long does it take the bolt to fall from the ceiling to the floor of the elevator?
b) What are the displacements of the bolt and the elevator floor in this time?

Homework Equations

constant acceleration (with acceleration equal to gravity at 9.81m/s/s)

The Attempt at a Solution

I managed to get the first part, with t = 1.08 s, but now I am confused with part B. I am not sure what the displacement of the elevator floor really is (I cannot picture it) and I assumed the displacement of the bolt at this time was the 3m that it fell (that's the value of displacement I used to get the time in the first place) SO now I am quite confused.

Can anyone give me a hand?
Thanks!

You can do the problem from either frame of reference ( the elevator - or - the building ). They're both inertial frames of reference. But I think the second part of the problem is written with the idea that the building is the frame of reference.

During the time it takes the bolt to fall (1.08s), how far does the floor of the elevator move? - and in what direction?

So, the bolt fell 3m minus the amount the floor moved upward - both in that 1.08 s of time. (BTW: I would expect up to be positive & down to be negative - for your final answers.)

I'll do a follow-up about why you got the right answer for A even though you ignored the displacement of the floor.

 

[cal.queen92]

Actually, i don't know if I got the right answer for the time, and if you say I need to consider the displacement of the elevator, I'll try again, though I'm confused now how to find the time it takes for the bolt to hit the ground...

 

[rl.bhat]

Take floor as the refeence frame. Then the bolt is moving down wards direction with an initial velocity vo. Then the displacement, initial velocity and g are in the same direction.
Write the proper kinematic equation to solvce for t.

 

[SammyS]

If you use the elevator as the reference frame, then the displacement of the floor of the elevator is zero. I doubt that the answer key has that.

From the frame of reference of the building, the bolt has the same initial velocity as the elevator.

Let's say the bolt come loose at time t=0, and the position of the floor at that time is: y₀. Then the position of the bolt is y₀+3.0 at time t=0.

Position of the floor at time t: (Remember, the elevator & its floor have constant velocity v₀ = 2.5m/s .)

y_floor = y₀ + v₀·t .​

Position of the bolt at time t: (The initial velocity of the bolt is also v₀=2.5m/s at t=0. This is using the building as the reference frame.)

y_bolt = y₀ + 3.0 + v₀·t ‒ (1/2)g·t² .​

If you set y_floor = y_bolt, and solve for t, you get the time at which the bolt hits the floor. Most everything cancels and you get:

3.0 = (1/2)g·t², which is the same as if the bolt falls 3.0 m starting at rest.

BTW: I see that your answer to part (a) is incorrect.

 

[cal.queen92]

Okay, so to find the time it takes the bolt to fall to the ground I let the equation for the position of the floor equal the equation for the position of the bolt, and I end up with that final, smaller equation.

What I'm confused about is the direction of the velocity, acceleration, and 3.0m. Someone above said they all had the same, however, Would the acceleration be negative (gravity pushing down), the velocity be positive (moving up, opposing negative gravity) and the 3.0m be negative since it moves downward, in the same direction as gravity?

That's where I'm getting lost...

 

[cal.queen92]

I've drawn a sketch and think I understand now the direction everything is going and have ended up with t=0.78s

I figure, that if I take the acceleration as negative (since it is going down) as well as the 3.0m, then as i work through the equation, I end up with a positive time. (Velocity is positive as it opposes gravity - goes up):

3.0 = (-1/2)g*t^2 --> 3.0 = 4.81*t^2 etc, etc... Is this proper reasoning? Or not logical?

 

[cal.queen92]

Now I don't know what the initial y would be? to fill in the equations...

 

[rl.bhat]

The initial velocity of the bolt is 2.5 m/s in the upward direction.
After timr t, the bolt reaches the floor of the elevator. Let the bolt covers h1 and floor moves up through h2 during that time. Then
h1 = vo*t - 0.5*g*t^2
h2 = vo*t and h = h1 + h2.
Now solve for t.

 

[cal.queen92]

I used the instructions earlier to solve for t and got 0.78s... I think that's right?

Now i can't figure out how to use your equations to solve for time to see if I get he same answer twice. I don't have h1 or h2, and no t, so I am missing some values to solve for t aren't I?

And when I use the result for time I got with the previous equations, I just get 3.0m for the displacement of the bolt... (this is when I subtract the result for h2 from h1 - to take the displacement of the elevator away from that of the bolt)

 

[cal.queen92]

I used the instructions earlier to solve for t and got 0.78s... I think that's right?

Now i can't figure out how to use your equations to solve for time to see if I get he same answer twice. I don't have h1 or h2, and no t, so I am missing some values to solve for t aren't I?

And when I use the result for time I got with the previous equations, I just get 3.0m for the displacement of the bolt... (this is when I subtract the result for h2 from h1 - to take the displacement of the elevator away from that of the bolt)

 

[rl.bhat]

h1 = vo*t - 0.5*g*t^2...(1)
h2 = vo*t ...(2)
and h = h1 + h2= 3. Add eq.
3 = 2*2.5*t - 0.5*9.8*t^2
Solve for t.